On a generalized Erdős-Rademacher problem

Abstract

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $\lfloor n^2/4\rfloor +t$ edges and triangle covering number $s$, we determine (for large $n$) sharp bounds on the minimum number of triangles in $G$ and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erd\H os, and Lovász-Simonovits whose results apply only to $s\le t$. Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture.

arXiv comment21 pages, 1 figure

References in corpus (1)

• The number of triangles is more when they have no common vertex